For comparison, the classic Griffith model of brittle fracture
simplifies \(\mathcal {S}\)  to be constant, and a crack \(\Gamma _t\) 
to be predefined at the interface \(\Sigma _t\) .
This simplification results in a square-root singularity of the displacement \(u_t\) 
and infinite stress at the crack tip (front) \(\partial \Gamma _t\) .
This is the main disadvantage of the Griffith model,
we refer to [1]} for a discussion.
A model, consistent with the physics of quasi-brittle fracture for non-constant
\(\mathcal {S}\) , was suggested by Barenblatt [2]}.
It takes into account the surface cohesion from the meso-level such that
the interface surfaces close in a smooth way, and thus allow healing of the crack.
Indeed, after solving problem (REF ) according to Barenblatt,
the set of points where an opening \([\![u_t]\!]\ne 0\)  occurs, determines
the a-priori unknown crack \(\Gamma _t\)  along the interface \(\Sigma _t\) .
This is the complement to the closed part of the interface where \([\![u_t]\!]=0\) .
